# Algebraic topology – Calculation of the degree of a map

Let $$q: S ^ n rightarrow S ^ n$$ to be the map that quotients the lower hemisphere in the south pole$$s$$ . I am asked to calculate the degree of this map.

Yes $$D_ +$$ refers to the upper hemisphere and then $$q: D_ + rightarrow S ^ n – s$$ is a homeomorphism and so by the formula of the local degree, the degree is $$1$$ or $$-1$$.

I've tried to show that this card does not send any points to its antipode and that it is homotopic to the identity. So, he must have a degree $$1$$.

It's intuitively quite clear to me but I do not know how to write a formal proof.
Any idea / correction / help will be greatly appreciated.